拟相对论薛定谔方程基态解的存在性与爆破行为

Existence and Blow-Up Behavior of Ground State Solutions for Pseudo-Relativistic Schr?dinger Equations

针对如下约束极小化问题d_(a)(q)=inf{u∈H^(1/2)(R^(3)),∫_(R3)|u|^2dx=1}Eq(u),(0,1)其中E_(q)(u)为拟相对论薛定谔方程的能量泛函E_(q)(u)=1/2∫_(R3)u(√-△+m^(2)-m)udx-a/q+2∫_(R3)|u|^(q+2)dx.对任意q∈(0,2/3),该文证明了问题(0.1)至少存在一个径向对称的非负可达元;并在g↗2/3时,细致分析了可达元的爆破行为.

For the following constrained minimization problem d_(a)(q)=inf{u∈H^(1/2)(R^(3)),∫_(R3)|u|^2dx=1}Eq(u), where E_(q)(u)is the energy functional of the pseudo-relativistic Schrodinger equation E_(q)(u)=1/2∫_(R3)u(√-△+m^(2)-m)udx-a/q+2∫_(R3)|u|^(q+2)dx. For any q∈(0,2/3).the article proved that the problem(0.1)has at least one radially symmetric non-negative minimizer;and analyzed the blow-up behavior of the minimizer as q↗2/3.